Theoretically, there is no need to compute long segments of field lines to find a Poincaré plot. A basic result from dynamical systems theory states that  if \(\mathbf {B}\)  has a nowhere-vanishing toroidal component and \(\mathbf {B}\) -lines never leave \(Q\)  there is a smooth mapping \(\Phi :P\rightarrow P\)  that sends a point \(\mathbf {x}\in P\)  to that point's next intersection with \(P\) , \(\Phi (\mathbf {x})\) .
The mapping \(\Phi \)  is known as the first-return map, or the Poincaré map. Because \(\mathbf {B}\)  is divergence-free, \(\Phi \)  has the remarkable property that if \(U\subset P\)  is any region in \(P\)  then the magnetic flux though \(\Phi (U)\)  is the same as the magnetic flux through \(U\) , which says that \(\Phi \)  is a symplectic mapping relative to a symplectic form[1]} on \(P\)  determined by the magnetic flux.
